Forbidden subgraphs

Inclusions

The map shows the inclusions between the current class and a fixed set of landmark classes. Minimal/maximal is with respect
to the contents of ISGCI. Only references for direct inclusions are given. Where no reference is given, check equivalent classes
or use the Java application. To check relations other than inclusion (e.g. disjointness) use the Java application, as well.

Parameters

The acyclic chromatic number
of a graph $G$ is the smallest size of a vertex partition $\{V_1,\dots,V_l\}$ such that each $V_i$ is an independent set and
for all $i,j$ that graph $G[V_i\cup V_j]$ does not contain a cycle.

The bandwidth
of a graph $G$ is the
shortest maximum "length" of an edge over all one dimensional
layouts of $G$.
Formally, bandwidth
is defined as
$\min_{i \colon V \rightarrow \mathbb{N}\;}\{\max_{\{u,v\}\in E\;}
\{|i(u)-i(v)|\}\mid i\text{ is injective}\}$.

A book embedding of a graph $G$ is an embedding of $G$ on a collection of half-planes (called pages) having the same line
(called spine) as their boundary, such that the vertices all lie on the spine and there are no crossing edges. The book thickness
of a graph $G$ is the smallest number of pages over all book embeddings of $G$.

Consider the following decomposition of a graph $G$ which is defined
as a pair $(T,L)$ where $T$ is a binary tree and $L$ is a bijection
from $V(G)$ to the leaves of the tree $T$. The function
$\text{cut-bool} \colon 2^{V(G)} \rightarrow R$ is
defined as $\text{cut-bool}(A)$ := $\log_2|\{S \subseteq
V(G) \backslash A \mid \exists X \subseteq A \colon
S = (V(G) \backslash A) \cap \bigcup_{x \in X} N(x)\}|$.
Every edge $e$ in $T$
partitions the vertices $V(G)$ into $\{A_e,\overline{A_e}\}$ according
to the leaves of the two connected components of $T - e$.
The booleanwidth of the above decomposition $(T,L)$
is $\max_{e \in E(T)\;} \{ \text{cut-bool}(A_e)\}$.
The booleanwidth
of a graph $G$ is the
minimum booleanwidth of a decomposition of $G$ as above.

A branch decomposition of a graph $G$ is a pair $(T,\chi)$,
where $T$ is a binary tree and $\chi$ is a bijection, mapping leaves
of $T$ to edges of $G$. Any edge $\{u, v\}$ of the tree divides
the tree into two components and
divides the set of edges of $G$ into two parts $X, E \backslash X$,
consisting of edges mapped to the leaves of each component. The
width of the edge $\{u,v\}$ is the number of vertices of $G$ that is incident both with an edge in
$X$ and with an edge in $E \backslash X$. The width of the
decomposition $(T,\chi)$ is the maximum width of its edges. The
branchwidth
of the graph $G$ is the
minimum width over all branch-decompositions of $G$.

Consider a decomposition $(T,\chi)$ of a graph $G$ where $T$
is a binary tree with $|V(G)|$ leaves and $\chi$ is a
bijection mapping the leaves of $T$ to the vertices of
$G$. Every edge $e \in E(T)$ of the tree $T$ partitions the
vertices of the graph $G$ into two parts $V_e$ and $V
\backslash V_e$ according to the leaves of the two connected
components in $T - e$. The width of an edge $e$ of the tree
is the number of edges of a graph $G$ that have exactly one
endpoint in $V_e$ and another endpoint in $V \backslash V_e$.
The width of the decomposition $(T,\chi)$ is the largest
width over all edges of the tree $T$. The
carvingwidth
of a graph is the
minimum width over all decompositions as above.

Unbounded from Graph isomorphism assuming Polynomial,GI-complete disjoint.Unbounded from acyclic chromatic numberUnbounded from branchwidthUnbounded from book thicknessUnbounded from maximum cliqueUnbounded from Weighted clique assuming Polynomial,NP-complete disjoint.Unbounded from rankwidthUnbounded from Clique assuming Polynomial,NP-complete disjoint.Unbounded from degeneracyUnbounded from cliquewidthUnbounded from treewidthUnbounded from chromatic numberUnbounded from booleanwidth

The cutwidth of a graph $G$ is the smallest integer $k$ such
that the vertices of $G$ can be arranged in a linear layout
$v_1, \ldots, v_n$ in such a way that for every $i = 1,
\ldots,n - 1$, there are at most $k$ edges with one endpoint
in $\{v_1, \ldots, v_i\}$ and the other in ${v_{i+1}, \ldots,
v_n\}$.

Unbounded from pathwidthUnbounded from degeneracyUnbounded from book thicknessUnbounded from bandwidthUnbounded from carvingwidthUnbounded from treewidthUnbounded from Graph isomorphism assuming Polynomial,GI-complete disjoint.Unbounded from acyclic chromatic numberUnbounded from Clique assuming Polynomial,NP-complete disjoint.Unbounded from maximum cliqueUnbounded from chromatic numberUnbounded from booleanwidthUnbounded from Weighted clique assuming Polynomial,NP-complete disjoint.Unbounded from rankwidthUnbounded from maximum degreeUnbounded from cliquewidthUnbounded from branchwidth

Unbounded from maximum induced matchingUnbounded from diameterUnbounded from distance to co-clusterUnbounded from distance to blockUnbounded from distance to cographUnbounded from distance to clusterUnbounded from minimum clique coverUnbounded from minimum dominating setUnbounded from maximum independent set

Unbounded from distance to outerplanarUnbounded from Weighted clique assuming Polynomial,NP-complete disjoint.Unbounded from chromatic numberUnbounded from Graph isomorphism assuming Polynomial,GI-complete disjoint.Unbounded from acyclic chromatic numberUnbounded from maximum cliqueUnbounded from branchwidthUnbounded from degeneracyUnbounded from treewidthUnbounded from Clique assuming Polynomial,NP-complete disjoint.Unbounded from distance to blockUnbounded from booleanwidthUnbounded from cliquewidthUnbounded from rankwidthUnbounded from book thicknessUnbounded from pathwidth

Unbounded from branchwidthUnbounded from chromatic numberUnbounded from Clique assuming Polynomial,NP-complete disjoint.Unbounded from book thicknessUnbounded from degeneracyUnbounded from rankwidthUnbounded from Graph isomorphism assuming Polynomial,GI-complete disjoint.Unbounded from acyclic chromatic numberUnbounded from Weighted clique assuming Polynomial,NP-complete disjoint.Unbounded from booleanwidthUnbounded from maximum cliqueUnbounded from cliquewidthUnbounded from treewidth

Unbounded from maximum cliqueUnbounded from Weighted clique assuming Polynomial,NP-complete disjoint.Unbounded from chromatic numberUnbounded from degeneracyUnbounded from Clique assuming Polynomial,NP-complete disjoint.Unbounded from book thicknessUnbounded from acyclic chromatic number

For a graph $G = (V,E)$ an induced matching is an edge subset
$M \subseteq E$ that satisfies the following two conditions:
$M$ is a matching of the graph $G$ and there is no edge in $E
\backslash M$ connecting any two vertices belonging to edges
of the matching $M$. The parameter
maximum induced matching
of a graph $G$ is the
largest size of an induced matching in $G$.

Unbounded from tree depthUnbounded from pathwidthUnbounded from treewidthUnbounded from acyclic chromatic numberUnbounded from chromatic numberUnbounded from degeneracyUnbounded from distance to co-clusterUnbounded from Weighted clique assuming Polynomial,NP-complete disjoint.Unbounded from book thicknessUnbounded from vertex coverUnbounded from distance to clusterUnbounded from cliquewidthUnbounded from booleanwidthUnbounded from maximum cliqueUnbounded from distance to cographUnbounded from branchwidthUnbounded from distance to blockUnbounded from Graph isomorphism assuming Polynomial,GI-complete disjoint.Unbounded from diameterUnbounded from Clique assuming Polynomial,NP-complete disjoint.Unbounded from rankwidthUnbounded from maximum induced matching

Unbounded from treewidthUnbounded from pathwidthUnbounded from book thicknessUnbounded from cliquewidthUnbounded from Clique assuming Polynomial,NP-complete disjoint.Unbounded from maximum degreeUnbounded from distance to blockUnbounded from distance to linear forestUnbounded from branchwidthUnbounded from booleanwidthUnbounded from degeneracyUnbounded from acyclic chromatic numberUnbounded from Graph isomorphism assuming Polynomial,GI-complete disjoint.Unbounded from Weighted clique assuming Polynomial,NP-complete disjoint.Unbounded from maximum cliqueUnbounded from rankwidthUnbounded from bandwidthUnbounded from distance to outerplanarUnbounded from chromatic number

A clique cover of a graph $G = (V, E)$ is a partition $P$ of $V$
such that each part in $P$ induces a clique in $G$. The
minimum clique cover
of $G$ is the minimum number of
parts in a clique cover
of $G$. Note that the clique cover number of
a graph is exactly the chromatic number of its complement.

A dominating set of a graph $G$ is a subset $D$ of its vertices, such
that every vertex not in $D$ is adjacent to at least one member of
$D$. The parameter minimum dominating set
for graph
$G$ is the minimum number of vertices in a dominating set for $G$.

Unbounded from booleanwidthUnbounded from maximum cliqueUnbounded from Weighted clique assuming Polynomial,NP-complete disjoint.Unbounded from treewidthUnbounded from book thicknessUnbounded from cliquewidthUnbounded from acyclic chromatic numberUnbounded from degeneracyUnbounded from Clique assuming Polynomial,NP-complete disjoint.Unbounded from chromatic numberUnbounded from Graph isomorphism assuming Polynomial,GI-complete disjoint.Unbounded from rankwidthUnbounded from branchwidth

Let $M$ be the $|V| \times |V|$ adjacency matrix of a graph $G$. The
cut rank of a set $A \subseteq V(G)$ is the rank of the submatrix of
$M$ induced by the rows of $A$ and the columns of $V(G) \backslash A$.
A rank decomposition of a graph $G$ is a pair $(T,L)$ where $T$ is a
binary tree and $L$ is a bijection from $V(G)$ to the leaves of the
tree $T$. Any edge $e$ in the tree $T$ splits $V(G)$ into two parts
$A_e, B_e$ corresponding to the leaves of the two connected components
of $T - e$. The width of an edge $e \in E(T)$ is the cutrank of $A_e$.
The width of the rank-decomposition $(T,L)$ is the maximum width of an
edge in $T$. The rankwidth
of the graph
$G$ is the minimum width of a rank-decomposition of $G$.

A tree depth decomposition of a graph $G = (V,E)$ is a rooted
tree $T$ with the same vertices $V$, such that, for every
edge $\{u,v\} \in E$, either $u$ is an ancestor of $v$ or $v$
is an ancestor of $u$ in the tree $T$. The depth of $T$ is
the maximum number of vertices on a path from the root to any
leaf. The tree depth
of a graph
$G$ is the minimum depth among all tree depth decompositions.

Unbounded from Clique assuming Polynomial,NP-complete disjoint.Unbounded from Weighted clique assuming Polynomial,NP-complete disjoint.Unbounded from maximum cliqueUnbounded from degeneracyUnbounded from branchwidthUnbounded from diameterUnbounded from treewidthUnbounded from acyclic chromatic numberUnbounded from rankwidthUnbounded from chromatic numberUnbounded from booleanwidthUnbounded from book thicknessUnbounded from pathwidthUnbounded from Graph isomorphism assuming Polynomial,GI-complete disjoint.Unbounded from cliquewidth

Unbounded from booleanwidthUnbounded from maximum cliqueUnbounded from maximum induced matchingUnbounded from tree depthUnbounded from diameterUnbounded from acyclic chromatic numberUnbounded from distance to blockUnbounded from distance to co-clusterUnbounded from treewidthUnbounded from cliquewidthUnbounded from book thicknessUnbounded from Weighted clique assuming Polynomial,NP-complete disjoint.Unbounded from branchwidthUnbounded from distance to clusterUnbounded from maximum matchingUnbounded from degeneracyUnbounded from distance to cographUnbounded from Clique assuming Polynomial,NP-complete disjoint.Unbounded from rankwidthUnbounded from chromatic numberUnbounded from Graph isomorphism assuming Polynomial,GI-complete disjoint.Unbounded from pathwidth

Problems

Problems in italics have no summary page and are only listed when
ISGCI contains a result for the current class.